(expires 11 March)
Find all the solutions to the differential equation

Among them, single out the one for which .
[Hint: read the help page for dsolve, or just do it in your
head. It is that easy.]

14.

(expires 11 March)
Have Maple find analytic solutions to the following system of
differential equations,

with initial conditions:
. Let us denote
the solutions by
(since they depend on the parameter
).

For taking all integer values from -10 to 10, and , plot the functions in blue, and the functions
in red, all on the same graph. (Yes, you will then have 42 functions
plotted on the same graph.)
[This is certainly a case when you don't want to retype the
functions that Maple finds.
You will almost certainly need to read the help page for dsolve.
I also found subs, unapply, and seq useful.]

15.

(expires 11 March)
For the functions and found in problem #14, plot
the parametric curves
for integer values of between
and and on the same graph. Use the
view
option of plot to only show what lies in the region
, and use a sequence of colors so that each solution is a
different color. [ HINT: you might find something like
seq(COLOR(HUE,i/11),i=0..10) useful for the latter.]

16.

(expires 11 March)
Find all the fixed points of the system

a fixed point being a solution for which both and
stay constant. For each of these points, describe the behavior of the
solutions that have initial conditions nearby. You can use Maple to
figure out what happens for nearby points, or you can use more
mathematical methods.

NOTE:The fact that there are various notations for
differential equations is purely intentional.